Mrho=0.77; Mpi=0.139;

al1=1/(2(Mrho^2-Mpi^2))

al0=1/2 - al1 Mpi^2

MM[n_]=Sqrt[1/al1 (n-al0)];

M=Table[Sqrt[1/al1 (n-al0)],{n,2,4}]

R[n_,al_]=Gamma[al0+al1 t+n]

-1/(b-1)+Sum[(a+n-1)!/(a-1)! 1/n! 1/(n+1-b),{n,1,Infinity}] // FullSimplify

Original[a_,b_]=Gamma[1-b] Gamma[1-a]/Gamma[1-a-b]

Res1[a_,b_]=Sum[Gamma[n+a]/(Gamma[n] Gamma[a]) 1/(b-n),{n,1,Infinity}] // FullSimplify

Original[a,b]/Res1[a,b] // FullSimplify

Res2[a_,b_]=-(a+b-1) Sum[Binomial[a+n-1,n] 1/(n+1-b),{n,0,Infinity}] // FullSimplify

Original[a,b]/Res2[a,b] // FullSimplify

Table[MM[n],{n,2,8}]

als=a0+a1 s; alt=a0+a1 t; alu=a0+al u;

Phi[al_,be_]=Gamma[1-al]Gamma[1-be]/Gamma[1-al-be]

A[s_,t_,u_]=-lam/2(Phi[als,alt]+Phi[als,alu]-Phi[alt,alu]) // Simplify;

T0[s_,t_,u_]=3 A[s,t,u] + A[t,u,s]+A[u,s,t] // FullSimplify;

T1[s_,t_,u_]= A[t,u,s]+A[u,s,t] // FullSimplify;

T2[s_,t_,u_]= A[t,u,s]-A[u,s,t] // FullSimplify;

T2n[s_,t_,u_]=- lam Phi[alt,alu];

T0n[s_,t_,u_]=-lam/2 (3 Phi[als,alt]+3 Phi[als,alu] - Phi[alt,alu]) ;

T1n[s_,t_,u_]=-lam( Phi[als,alt]- Phi[als,alu] ) ;

rnn[n_]= n/(2^n (2n+1)!!) al1^n (MM[n]^2-4 Mpi^2)^n;

Gammann[n_]=0.728 rnn[n] Sqrt[MM[n]^2-4 Mpi^2]/(al1 MM[n]^2);

Gammann[2]*3*1000 // N

Gammann[3]*2*1000 // N

Gammann[4]*3*1000 // N

Clear[RR, rnl, alt]

a0=al0; a1=al1;

alt[t_]=a0+a1 t

RR[n_,t_]:=Gamma[alt[t]+n]/(Gamma[alt[t]]*Gamma[n]);

rnl[n_,l_]:=1/2 NIntegrate[RR[n,1/2(4 Mpi^2 -MM[n]^2)(1-z)] LegendreP[l,z],{z,-1,1}]

rnl[1,1] 
rnn[1]
rnl[2,2]
rnn[2]

rnl[3,3] // N
rnn[3]

{Gammann[2]*3000,Gammann[3]*2000,Gammann[4]*3000,Gammann[5]*2000,Gammann[6]*3000,Gammann[7]*2000,Gammann[8]*3000} // N

Clear[Gammanl]

Gammanl[n_,l_]:=0.728 rnl[n,l] Sqrt[MM[n]^2-4 Mpi^2]/(al1 MM[n]^2);

{Gammanl[1,0]*3000,Gammanl[2,1]*2000,Gammanl[3,2]*3000,Gammanl[4,3]*2000} // N